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Metrix MO 2020 8p

geometry problems from Metrix Mathematical Olympiad 2020 and the Shortlist (unused problems) with aops links


2020
first posted here
collected inside aops here

contest problems

Does there exist a heptagon $P$ satisfying the following:
There does not exist a square with exactly three of the vertices on the boundary of $P$.

 by MNJ2357
$ABC$ be a scalene triangle. Let $D$ be a point on $\overline{BC}$ such that $\overline{AD}$ is the angle bisector of $\angle BAC$. Let $M$ be the midpoint of $\widehat{BAC}$ of the circumcircle of $\odot(ABC)$. Let $\overline{DN}\perp\overline{BC}$ where $N$ lies on $\overline{AM}$. $K$ be a point on segment $NM$ such that $3KM=KN$. Let $P,Q$ be two points on $\overline{AM}$ such that $KP=KQ$ and $BCPQ$ lie on a circle $\omega$. Show that the reflection of $D$ over $A$ lies on $\omega$.

by Amar_04
sample problems

An acute triangle $ABC$ ($AB\neq AC$) with circumcircle $\omega$ is given, and $M$ is the midpoint of arc  BAC . Suppose $D$ is the point on segment $BC$ such that the circumcircle of $MAD$ is tangent to $BC$, and suppose $P$ is the point on segment $BC$ such that $AP$ and $MD$ concur on $\omega$. Let $Q$ be the intersection of $MA$ and $BC$, $N$ be the midpoint of $MD$, and $X$ be the intersection of $NQ$ and $MP$. Prove that $\angle MDB=\angle XDC$.

by MNJ2357
The point $H$ is a orthocenter of the triangle $ABC$ , given the conditions that $\odot(ABH)\cap HC=T,\odot(ACH)\cap HB=K$, $\odot(KTA)\cap \odot(ABC)=N$ , $AH\cap \odot(HTK)=P,\odot(HTK)\cap \odot(PNA)=Q$
$\odot(HBC)\cap HQ=M$ with $HACB'$-cyclic, $HABC'$-cyclic. $\angle BAH=\angle CC'A,\angle CAH=\angle BB'A$
$\odot(ABB')\cap \odot(ACC')=H'$ .Then prove that $H',B',C',M$-cyclic

by Functional_equation

unused problems

Let $ABC$ be a triangle with centroid $G$. $GD$, $GE$, $GF$ are symmedians of triangles $GBC$, $GCA$, $GAB$, respectively. $L$ is symmedian point of $ABC$. $L^*$ is inversion of $L$ in circle $(ABC)$. Prove that $AD$, $BE$, $CF$ and $GL^*$ are concurrent.

by buratinogiggle

Let the points $H$ and $N_9$ to be the orthocenter and the Nine-Point center of $\vartriangle ABC$ respectively and let $R$ to be the circumradius of $\vartriangle ABC$. If $\angle BAC = \alpha$ , $\angle ABC = \beta$, $\angle ACB = \gamma$. Prove that$$\left( \frac{2HN_9}{R}\right)^2 \le  9 - 8\sqrt3 \sin \alpha \cdot \sin \beta \cdot \sin \gamma.$$
 by Functional_equation

Let $D, E,$ and $F$ be the respective feet of the $A, B,$ and $C$ altitudes in $\triangle ABC$, and let $M$ and $N$ be the respecive midpoints of $\overline{AC}$ and $\overline{AB}$. Lines $DF$ and $DE$ intersect the line through $A$ parallel to $BC$ at $X$ and $Y$, respectively. Lines $MX$ and $YN$ intersect at $Z$. Prove that the circumcircles of $\triangle EFZ$ and $\triangle XYZ$ are tangent.

by mathman3880
In triangle $ABC$ denote $I_A$ as the center of the excircle wrt $A$. Let the excircle touch the sides $AB, BC, CA$ at $M, N, P$ respectivly. Let $T$ be the midpoint of $MP$ , $TC$ and $MN$ hit at $T'$ and $BC$ , $PM$ hit at $T''$ . Prove $I_A$ is the orthocenter of $TT'T''.$

by Mr.C

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